We use homogenization theory to investigate the asymptoticmacrodispersion in arbitrary nonuniform velocity fields, which showsmall-scale fluctuations. In the first part of the paper, amultiple-scale expansion analysis is performed to study transportphenomena in the asymptotic limit epsilon << 1, where epsilonrepresents the ratio between typical lengths of the small and largescale. In this limit the effects of small-scale velocity fluctuationson the transport behavior are described by a macrodispersive term, andour analysis provides an additional local equation that allowscalculating the macrodispersive tensor. For Darcian flow fields we showthat the macrodispersivity is a fourth-rank tensor. Ifdispersion/diffusion can be neglected, it depends only on the directionof the mean flow with respect to the principal axes of anisotropy ofthe medium. Hence the macrodispersivity represents a medium property.In the second part of the paper, we heuristically extend the theory tofinite epsilon effects. Our results differ from those obtained in thecommon probabilistic approach employing ensemble averages. Thisdemonstrates that standard ensemble averaging does not consistentlyaccount for finite scale effects: it tends to overestimate thedispersion coefficient in the single realization.